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Evaluate the following without using a calculator.

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A patient-controlled analgesia (PCA) contains $\\var{m}$ {unit} of a certain drug made in $\\var{v}$ mL of IV fluid. The machine is set to deliver a bolus dose of $\\var{b}$ mL each time the button is pressed. If the button is pressed $\\var{n}$ times in one hour, how much of the drug has been received in that hour?

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[[0]] {unit}

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First work out the concentration

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\\begin{align}\\text{concentration (mg/mL)}&=\\frac{\\text{mass (mg)}}{\\text{volume (mL)}}\\\\&=\\frac{\\var{m} \\text{ mg}}{\\var{v} \\text{ mL}}\\\\&=\\var{c} \\text{ mg/mL}\\end{align}

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\\begin{align}\\text{concentration (mcg/mL)}&=\\frac{\\text{mass (mcg)}}{\\text{volume (mL)}}\\\\&=\\frac{\\var{m} \\text{ mcg}}{\\var{v} \\text{ mL}}\\\\&=\\var{c} \\text{ mcg/mL}\\end{align}

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Now the individual dose:

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\\begin{align}\\text{individual dose (mg)}&=\\text{concentration (mg/mL)}\\times \\text{volume (mL)}\\\\&=\\var{c} \\text{ mg/mL}\\times \\var{b} \\text{ mL}\\\\&=\\var{ans} \\text{ mg}\\end{align}

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\\begin{align}\\text{individual dose (mcg)}&=\\text{concentration (mcg/mL)}\\times \\text{volume (mL)}\\\\&=\\var{c} \\text{ mcg/mL}\\times \\var{b} \\text{ mL}\\\\&=\\var{ans} \\text{ mcg}\\end{align}

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Now since the patient received $\\var{n}$ individual doses, they received:

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\\begin{align}\\text{total amount (mg)}&=\\text{individual dose (mg)}\\times \\text{number of doses}\\\\&=\\var{ans}\\text{ mg} \\times \\var{n}\\\\&=\\var{ans2} \\text{ mg}\\end{align}

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\\begin{align}\\text{total amount (mcg)}&=\\text{individual dose (mcg)}\\times \\text{number of doses}\\\\&=\\var{ans}\\text{ mcg} \\times \\var{n}\\\\&=\\var{ans2} \\text{ mcg}\\end{align}

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